Homogenization of the eigenvalues of the Neumann-Poincar\'e operator

In this article, we investigate the spectrum of the Neumann-Poincar\'e operator associated to a periodic distribution of small inclusions with size $\varepsilon$, and its asymptotic behavior as the parameter $\varepsilon$ vanishes. Combining techniques pertaining to the fields of homogenization and potential theory, we prove that the limit spectrum is composed of the `trivial' eigenvalues $0$ and $1$, and of a subset which stays bounded away from $0$ and $1$ uniformly with respect to $\varepsilon$. This non trivial part is the reunion of the \textit{Bloch spectrum}, accounting for the collective resonances between collections of inclusions, and of the \textit{boundary layer spectrum}, associated to eigenfunctions which spend a not too small part of their energies near the boundary of the macroscopic device. These results shed new light about the homogenization of the voltage potential $u_\varepsilon$ caused by a given source in a medium composed of a periodic distribution of small inclusions with an arbitrary (possible negative) conductivity $a$, surrounded by a dielectric medium, with unit conductivity. In particular, we prove that the limit behavior of $u_\varepsilon$ is strongly related to the (possibly ill-defined) homogenized diffusion matrix predicted by the homogenization theory in the standard elliptic case. Additionally, we prove that the homogenization of $u_\varepsilon$ is always possible when $a$ is either positive, or negative with a `small' or `large' modulus.